The Quasigeostrophic Assumption
and the Inertial-Advective Wind
The quasigeostrophic omega and height tendency equations can be
derived directly from the equations of horizontal motion. Going
backwards from these two equations without removing the
synoptic-scaling provides some insight to what “quasigeostrophic”
means.
The two equations of horizontal motion in Cartesian coordinates
are
du
z
 g  fv
dt
x
(1)
x component

dv
z
 g  fu
dt
y
(2)
y component

The u-component of the geostrophic wind is obtained from (2)
z
ug   g f
y
(3)
 by f and substitute (3) into the
divide both sides of equation (2)
result.
dv
 f (ug  u)
dt

(4)
This equation states that there will be northward or southward
acceleration of the air parcel if the real wind differs from the
geostrophic wind.
Let’s assume that we have initially only a west wind (a jet stream
in the upper troposphere that lies along a line of latitude). For the
sake of argument,let’s also assume that there are no west-east
accelerations, so that (1) is zero, and the equation of horizontal
motion reduces to the geostrophic wind (for equation (1) and and
an acceleration given by (4).
The real wind can always be broken into a geostrophic and an
ageostrophic component
u = u g + ua
v = v g + va
(5a)
(5b)
Substitute (5a) into (4)
dv
  fua  f (ug  u)
dt
(6)
Equation (6) gives the ageostrophic wind that occurs if the wind is
 balance. Note that the far right hand side says
not in geostrophic
that subgeostrophic flow will be associated with northward
accelerations and vice versa.
The quasigeostrophic assumption can be applied to (6) if we try to
”retain” some of the acceleration to the left hand side of the
equation that is assumed to be zero in geostrophic flow. (Note: if
the left hand side is zero, then (6) becomes the geostrohpic wind
equation again.) To do that, we replace V in the equation of
motion on the left hand side with the geostrophic wind
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(essesntially, assuming that the ageostrophic wind along the y axis
is small).
Next, expand out the Lagrangian derivative
Vg
dV Vg

 Vg  Vg  w
dt
t
z

(7)
By replacing the real wind with the geostrophic wind on the left
hand side, we’re saying that the geostrophic wind “almost” is the
same as the real wind. This allows us to add back a bit of the
acceleration (which we have found out is related to divergence)
that the geostrophic assumption strictly does not allow (since,
except for the effect of the northward variation of the Coriolis
parameter, the geostrophic wind is non-divergent).
The first term to the right of the equals sign in (7) is the local
change of the geostrophic wind. This is due to changes in pressure
gradients (in turn due to isallobaric effects), and the motion of
troughs and ridges, for example. If we start with the assumptions
that the wind flow is zonal, this term is small or zero.  In
essence, this term captures the proportion of the ageostrophic flow
that is isallobaric.
The far right hand term, called the inertial-convective term, is zero
since w is zero in the upper troposphere (we'll look at the concept
of this term another time).
The second term to the right of the equals sign is essentially the
advection of the geostrophic wind by itself. In other words, as an
air parcel in geostrophic balance moves into a region with a
different pressure gradient, it will initially have its initial speed
which will be out of balance with the pressure gradient. (This is
3
sort of like a ball rolling down an inclined plane to a flat surface,
proceeding along the flat surface, where there is no gradient, by its
own momentum). This is called the "inertial advective wind"
With these assumptions, equation (7) becomes
dV
 Vg  Vg
dt
(8)
Putting (8) back into (6) (since there is no du/dt) we get

dv
Vg  Vg 
  fua  f (ug  u)
dt
(9)
We’ll now apply this to the real atmosphere by taking your first
look at jet streak dynamics.

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